Just yesterday, I had the opportunity to explain how I have handled similar situations in the past.
Please skip the parts of this copy/paste e-mail reply, but pay attention to the portion referring to the 12/12 front tie in to the low slope rear roof section and how I have trained my guys to install the Shingle Vent II and adapt it to work in the exact same type of scenario.
**I have used the Shingle Vent II Ridge vent on 2 1/2 per 12 pitch roofs, but I am not confident that someone else would take the appropriate due caution and additional measures to ensure it’s water-tight integrity.
If that option appeals to you, then this is what I do in those scenarios, if you have ever seen a cape cod 12/12 front of the house with 2/12 near flat roof dormer in the rear.
I install an additional baffle "Inside of where the center of the Ridge Vent product gets positioned. It really is just a piece of Aluminum J-Channel, with the longer extension being nailed to the flat roof and then that part of the J-Channel get either the flat roof membrane modified bitumen installed over it or shingles nailed up to the J feature. The J Profile faces down towards the eave. On your roof, this would be on both sides of the cut-out slot.
Then, I position the Shingle Vent II over the J-Channels, but prior to nailing it in place with 2 1/2" to 3" hand nails, I apply a very liberal amount of either Geocel or OSI NP-1 caulking sealant under where the Ridge Vent product is to make contact with the roof.
The interior J-Channel should prevent any blown in snow or rain. I have been using this technique since 1991 and have not had a call back so far, with between 15-25 such type situations.**
Otherwise, dependent upon your soffit width on the overhangs on the gable/rake edges, I would consider a powered gable vent. One on one side with the fan and one on the opposing wall, without the fan, so that one will pull the air flowage directly across, under the ridge, so one would be the intake and one would be the exhaust.
I noticed from your diagram, that you currently have small box shaped gable vents in those walls now and I wonder if the Attic Ventilation is currently being “Short-Circuited”, as the theory states as a potential affect. I doubt it in your case, even though they should have been sealed off to prevent that possibility from occurring.
I more than likely think that some poor workmanship caused the problems. When anyone removes any of the materials on your roof from either the old mushroom vent holes or the ridge vent slot covering, be on the roof and take pictures and feel if and where any of the wood is wet. If it is “Moist” continuously, it more than likely, but not assuredly, was from humidity building up the internal RH factor to a point of condensation, sufficient enough to accumulate and cause leaks. Once again, I doubt that to be the case, because I believe you stated that the leaks were during this current wave of 72 hour continuous rain we all had. We had around 10 1/2 inches in Carpentersville and Algonquin this past weekend. But, that also increases the humidity levels of the interior of improperly ventilated attics, so just one more item to be cognizant about.
If the wood is primarily wet where exactly the leak sounds came from, then it is just plain old faulty sealing of the products to the roof and inexperience, but that is how someone gets experienced, unfortunately.